There are two vessels of equal capacity, one full of pure milk and the second one-third full of water. The second vessel is then filled up out of the first, the contents of this second are then poured back into the first till it is full. Now the ratio of quantities of milk and water in the first vessel is $$a \colon b$$, where $$a$$ and $$b$$ are coprime natural numbers, then $$a + b = $$ ______.

Take each vessel to hold 3 units. The second vessel has 1 unit of water and receives 2 units of milk from the first, leaving 1 unit of milk in the first vessel, while the second holds milk and water in the ratio 2 to 1. Pouring back 2 units to fill the first vessel adds $$\frac{4}{3}$$ of milk and $$\frac{2}{3}$$ of water, so the first vessel has $$\frac{7}{3}$$ milk and $$\frac{2}{3}$$ water, a ratio of 7 to 2. Hence $$a + b = 9$$.